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Wavelet Transform
Eva Hostalkova
WAVELET TRANSFORM
Eva Hostalkova
Dept of Computing and Control Engineering
Institute of Chemical Technology, Prague
ATHENS Nov 2009
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 1 / 36
Wavelet Transform
Eva Hostalkova
Table of Contents
1 Introduction to Wavelet Transform2 Fourier Transform
Continuous Fourier TransformShort-Time Fourier Transform (STFT)
3 Wavelet TransformMulti-Resolution Analysis (MRA)Continuous Wavelet Transform (CWT)Wavelet Functions Properties
4 Discrete Wavelet TransformDiscrete and Continuous Wavelet TransformSubband Coding AlgorithmMatrix Interpretation
5 Wavelet Transform ApplicationsDiscontinuity Detection in the ECG SignalImage CompressionImage SegmentationNoise Reduction by Wavelet Shrinkage
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 2 / 36
Wavelet Transform
Eva Hostalkova
Table of Contents
1 Introduction to Wavelet Transform2 Fourier Transform
Continuous Fourier TransformShort-Time Fourier Transform (STFT)
3 Wavelet TransformMulti-Resolution Analysis (MRA)Continuous Wavelet Transform (CWT)Wavelet Functions Properties
4 Discrete Wavelet TransformDiscrete and Continuous Wavelet TransformSubband Coding AlgorithmMatrix Interpretation
5 Wavelet Transform ApplicationsDiscontinuity Detection in the ECG SignalImage CompressionImage SegmentationNoise Reduction by Wavelet Shrinkage
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 2 / 36
Wavelet Transform
Eva Hostalkova
Table of Contents
1 Introduction to Wavelet Transform2 Fourier Transform
Continuous Fourier TransformShort-Time Fourier Transform (STFT)
3 Wavelet TransformMulti-Resolution Analysis (MRA)Continuous Wavelet Transform (CWT)Wavelet Functions Properties
4 Discrete Wavelet TransformDiscrete and Continuous Wavelet TransformSubband Coding AlgorithmMatrix Interpretation
5 Wavelet Transform ApplicationsDiscontinuity Detection in the ECG SignalImage CompressionImage SegmentationNoise Reduction by Wavelet Shrinkage
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 2 / 36
Wavelet Transform
Eva Hostalkova
Table of Contents
1 Introduction to Wavelet Transform2 Fourier Transform
Continuous Fourier TransformShort-Time Fourier Transform (STFT)
3 Wavelet TransformMulti-Resolution Analysis (MRA)Continuous Wavelet Transform (CWT)Wavelet Functions Properties
4 Discrete Wavelet TransformDiscrete and Continuous Wavelet TransformSubband Coding AlgorithmMatrix Interpretation
5 Wavelet Transform ApplicationsDiscontinuity Detection in the ECG SignalImage CompressionImage SegmentationNoise Reduction by Wavelet Shrinkage
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 2 / 36
Wavelet Transform
Eva Hostalkova
Table of Contents
1 Introduction to Wavelet Transform2 Fourier Transform
Continuous Fourier TransformShort-Time Fourier Transform (STFT)
3 Wavelet TransformMulti-Resolution Analysis (MRA)Continuous Wavelet Transform (CWT)Wavelet Functions Properties
4 Discrete Wavelet TransformDiscrete and Continuous Wavelet TransformSubband Coding AlgorithmMatrix Interpretation
5 Wavelet Transform ApplicationsDiscontinuity Detection in the ECG SignalImage CompressionImage SegmentationNoise Reduction by Wavelet Shrinkage
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 2 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Introduction
Table of Contents
1 Introduction to Wavelet Transform
2 Fourier TransformContinuous Fourier TransformShort-Time Fourier Transform (STFT)
3 Wavelet TransformMulti-Resolution Analysis (MRA)Continuous Wavelet Transform (CWT)Wavelet Functions Properties
4 Discrete Wavelet TransformDiscrete and Continuous Wavelet TransformSubband Coding AlgorithmMatrix Interpretation
5 Wavelet Transform ApplicationsDiscontinuity Detection in the ECG SignalImage CompressionImage SegmentationNoise Reduction by Wavelet Shrinkage
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 3 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Introduction
Wavelet Transform (WT) History19th cent. - Jean B. Fourier: frequency analysis1909 - Alfred Haar: Haar function (not yet WT)since 1990s - WT related research and applications
Wavelet FunctionsWavelet = a small wave, i.e. an oscillatory function - zerooutside a bounded interval (having compact support)Some real, some complexDesigned as much smooth and symmetric as possibleSeldom analytic expression, mostly parametric (e.g.Morlet, Haar or Daubechies fcns)
10 20 30 40 50 60 70
−0.05
0
0.05
0.1
DAUBECHIES 2 WAVELET
time
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 4 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Introduction
Wavelet Transform (WT) History19th cent. - Jean B. Fourier: frequency analysis1909 - Alfred Haar: Haar function (not yet WT)since 1990s - WT related research and applications
Wavelet FunctionsWavelet = a small wave, i.e. an oscillatory function - zerooutside a bounded interval (having compact support)Some real, some complexDesigned as much smooth and symmetric as possibleSeldom analytic expression, mostly parametric (e.g.Morlet, Haar or Daubechies fcns)
10 20 30 40 50 60 70
−0.05
0
0.05
0.1
DAUBECHIES 2 WAVELET
time
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 4 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Introduction
WT AnalysisFor non-stationary signals (with time-varying frequencycontent)
WT Applications in Signal ProcessingNoise reductionDetection of trends and discontinuities in higherderivativesCompression (JPEG2000, FBI fingerprints database)Image edge detectionWatermarkingFeatures extraction for image segmentation
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 5 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Introduction
WT AnalysisFor non-stationary signals (with time-varying frequencycontent)
WT Applications in Signal ProcessingNoise reductionDetection of trends and discontinuities in higherderivativesCompression (JPEG2000, FBI fingerprints database)Image edge detectionWatermarkingFeatures extraction for image segmentation
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 5 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Fourier Transform
Table of Contents
1 Introduction to Wavelet Transform
2 Fourier TransformContinuous Fourier TransformShort-Time Fourier Transform (STFT)
3 Wavelet TransformMulti-Resolution Analysis (MRA)Continuous Wavelet Transform (CWT)Wavelet Functions Properties
4 Discrete Wavelet TransformDiscrete and Continuous Wavelet TransformSubband Coding AlgorithmMatrix Interpretation
5 Wavelet Transform ApplicationsDiscontinuity Detection in the ECG SignalImage CompressionImage SegmentationNoise Reduction by Wavelet Shrinkage
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 6 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Fourier Transform Continuous Fourier Transform
Continuous Fourier Transform (FT)Decomposes signals (real or complex) into the orthogonalbasis of complex exponentials e−j2πft of frequencies fFT for the angular frequency ω=2πf and time t:
X (ω) =
∫ +∞
−∞x(t) e−jωt dt (1)
where X (ω) is the Fourier transform of the function x(t)
Converges for a piece-wise smooth x(t) or finite energy:∫ ∞−∞|x(t)|2dt <∞ (2)
Inverse FT
x(t) =12π
∫ +∞
−∞X (ω) ejωt dω (3)
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 7 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Fourier Transform Continuous Fourier Transform
Continuous Fourier Transform (FT)Decomposes signals (real or complex) into the orthogonalbasis of complex exponentials e−j2πft of frequencies fFT for the angular frequency ω=2πf and time t:
X (ω) =
∫ +∞
−∞x(t) e−jωt dt (1)
where X (ω) is the Fourier transform of the function x(t)
Converges for a piece-wise smooth x(t) or finite energy:∫ ∞−∞|x(t)|2dt <∞ (2)
Inverse FT
x(t) =12π
∫ +∞
−∞X (ω) ejωt dω (3)
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 7 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Fourier Transform Continuous Fourier Transform
FT Analysis of Non-Stationary SignalsDiscrete FT for the discrete freq. index k =0, 1, . . . ,N−1and discrete time n (for the unit sampling frequency)
X (k) =N−1∑n=0
x(n) e−j2πkn/N (4)
0 100 200 300 400 500
−3
−2
−1
0
1
2
3
(a) SIGNAL 1
0 0.1 0.2 0.3 0.4
50
100
150
200
250 (b) MAGNITUDE SPECTRUM 1
0 100 200 300 400 500−1
−0.5
0
0.5
1
time
(c) SIGNAL 2
0 0.1 0.2 0.3 0.4
10
20
30
40
50
60
frequency
(d) MAGNITUDE SPECTRUM 2
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 8 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Fourier Transform STFT
Discrete Short-Time Fourier Transform (STFT)Non-stationary signal analysis (localizes frequencycomponents within time intervals)STFT of a discrete signal x(n) for k =0, 1, . . . ,N−1:
X (u, k) =N−1∑n=0
x(n) w(n − u)e−j2πkn/N (5)
where u denotes the window position
Uncertainty (Heisenberg) Principle
∆T ∆f = 1 (6)
∆f = Fs/N . . . frequency resolution (distance betweenadjacent spectral samples)∆T = N Ts . . . time resolution (window length)N . . . number of samples per windowFs = 1/Ts . . . sampling frequency
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 9 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Fourier Transform STFT
Discrete Short-Time Fourier Transform (STFT)Non-stationary signal analysis (localizes frequencycomponents within time intervals)STFT of a discrete signal x(n) for k =0, 1, . . . ,N−1:
X (u, k) =N−1∑n=0
x(n) w(n − u)e−j2πkn/N (5)
where u denotes the window position
Uncertainty (Heisenberg) Principle
∆T ∆f = 1 (6)
∆f = Fs/N . . . frequency resolution (distance betweenadjacent spectral samples)∆T = N Ts . . . time resolution (window length)N . . . number of samples per windowFs = 1/Ts . . . sampling frequency
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 9 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Fourier Transform STFT
STFT - Uncertainty Tradeoff
0 100 200 300 400 500−1
−0.5
0
0.5
1
time
(a) SIGNAL 2
510
15
1020
30
2
4
6
time u
(b) STFT: WINDOW L=32
frequency ξ
12
34
2040
6080
100120
10
20
30
time u
(c) STFT: WINDOW L=128
frequency ξ 1
1.5
2
50100
150200
250
10
20
30
time u
(d) STFT: WINDOW L=256
frequency ξ
(Using the Hamming window w - gently dropping to zero)Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 10 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Wavelet Transform
Table of Contents
1 Introduction to Wavelet Transform
2 Fourier TransformContinuous Fourier TransformShort-Time Fourier Transform (STFT)
3 Wavelet TransformMulti-Resolution Analysis (MRA)Continuous Wavelet Transform (CWT)Wavelet Functions Properties
4 Discrete Wavelet TransformDiscrete and Continuous Wavelet TransformSubband Coding AlgorithmMatrix Interpretation
5 Wavelet Transform ApplicationsDiscontinuity Detection in the ECG SignalImage CompressionImage SegmentationNoise Reduction by Wavelet Shrinkage
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 11 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Wavelet Transform MRA
Multi-Resolution Analysis (MRA)The wavelet transform deals with the uncertainty principleby MRAAnalyzing signals at different frequency bands withdifferent resolution:
Higher frequencies: good ∆T , poor ∆fLower frequencies: good ∆f , poor ∆T
Convenient for most real-world signals composed of:Long-lasting lower frequencies (approximations)Short-lasting higher frequencies (details - maininformation)
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 12 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Wavelet Transform CWT
Continuous Wavelet Transform (CWT)FT: correlation of a signal of finite energy with complexexponentials ej2πft of different frequencies fWT: correlation of a signal of finite energy with delatedand shifted versions ψu,s of the mother wavelet ψ:
ψu,s(t) =1√sψ
(t − u
s
)(7)
where u is the shift of ψ along the signal → timeand s is scale (dilation of ψ → the inverse of frequency)1/√
s ensures energy normalization
CWT of the signal x(t):
W ψx (s, u)=〈x , ψu,s〉 =
1√s
∫ +∞
−∞x(t)ψ∗
(t−u
s
)dt (8)
where * denotes the complex conjugate pairand 〈〉 stands for the inner product
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 13 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Wavelet Transform CWT
Continuous Wavelet Transform (CWT)FT: correlation of a signal of finite energy with complexexponentials ej2πft of different frequencies fWT: correlation of a signal of finite energy with delatedand shifted versions ψu,s of the mother wavelet ψ:
ψu,s(t) =1√sψ
(t − u
s
)(7)
where u is the shift of ψ along the signal → timeand s is scale (dilation of ψ → the inverse of frequency)1/√
s ensures energy normalization
CWT of the signal x(t):
W ψx (s, u)=〈x , ψu,s〉 =
1√s
∫ +∞
−∞x(t)ψ∗
(t−u
s
)dt (8)
where * denotes the complex conjugate pairand 〈〉 stands for the inner product
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 13 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Wavelet Transform CWT
CWT Computation Procedure (s =1)
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 14 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Wavelet Transform CWT
CWT Computation Procedure (s =5)
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 15 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Wavelet Transform CWT
CWT Properties
Linearity: implied by the properties of the inner product
Shift invariance:A shift in the signalalong the time axis7→ the equivalentshift of the waveletcoefficients withoutany changesDoes not apply tothe discrete WT(great drawback)
(a) Dual Tree CWT
Input
Wavelets
Level 1
Level 2
Level 3
Level 4
Scaling fn
Level 4
(b) Real DWT
Figure: Comparison of DWT and nearly invariant complex WT(CWT is even better - completely shift invariant)
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 16 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Wavelet Transform CWT
Orthogonality and orthonormality
Not necessary for wavelet analysis (biorthogonality formore freedom in wavelet filters construction)A set of shifted and dilated versions of the mother waveletψu,s , is orthonormal on the interval [a, b] when:
〈ψu,s , ψv ,r 〉 =
∫ b
aψu,s(t)ψ∗v ,r (t) dt =
{1 for u =v , s = r0 otherwise
(9)
A set of ψu,s , is orthogonal on the interval [a, b] when:
〈ψu,s , ψv ,r 〉 =
∫ b
aψu,s(t)ψ∗v ,r (t) dt =
{c for u =v , s = r0 otherwise
(10)where c is a constant, ∗ denotes a complex conjugate
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 17 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Wavelet Transform Wavelets Properties
Wavelet Functions Properties IAdmissibility condition∫ ∞
−∞
|Ψ(ω)|2
|ω|dω < +∞ (11)
where Ψ(ω) is the FT of ψ(t)Required for no loss of information during the analysis northe synthesisThe basis do not have to be orthogonal
Consequences of Admissibility ConditionOscillatory function (zero mean):∫ ∞
−∞ψ(t) dt = 0 (12)
Band-pass spectrum (Ψ(ω) vanishes at ω = 0):
|Ψ(ω)|2| ω=0 = 0 (13)
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 18 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Wavelet Transform Wavelets Properties
Wavelet Functions Properties IAdmissibility condition∫ ∞
−∞
|Ψ(ω)|2
|ω|dω < +∞ (11)
where Ψ(ω) is the FT of ψ(t)Required for no loss of information during the analysis northe synthesisThe basis do not have to be orthogonal
Consequences of Admissibility ConditionOscillatory function (zero mean):∫ ∞
−∞ψ(t) dt = 0 (12)
Band-pass spectrum (Ψ(ω) vanishes at ω = 0):
|Ψ(ω)|2| ω=0 = 0 (13)
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 18 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Wavelet Transform Wavelets Properties
Wavelet Functions Properties II
Time dilation of ψ causes shift and compression of themagnitude frequency spectrum |Ψ|
−50 0 50
−0.5
0
0.5
1DILATION OF THE WAVELET
s = 20
0 0.1 0.2 0.3 0.4 0.5
2
4
6
8
10
norm. frequency
FT OF THE DILATED WAVELET
NEWLAND WAVELET:
ψ(t) =1
j π2 t
(ej π t− ej π2 t)
WAVELET SCALING & DILATION:
ψu,s(t) =1√sψ
(t− u
s
)
−50 0 50
−0.5
0
0.5
1
s = 21
−50 0 50
−0.5
0
0.5
1
time
s = 22
s=20
s=21
s=22
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 19 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
Wavelet Transform Wavelets Properties
Desired Wavelet Functions Properties
Compact support (non-zero only on a restricted interval) -not necessaryRegularity:
Smoothness of ψ and vanishing of |Ψ(ω)| for large ω(small scales)Vanishing moments concept (see literature)
Symmetry (liner phase)
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 20 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
DWT
Table of Contents
1 Introduction to Wavelet Transform
2 Fourier TransformContinuous Fourier TransformShort-Time Fourier Transform (STFT)
3 Wavelet TransformMulti-Resolution Analysis (MRA)Continuous Wavelet Transform (CWT)Wavelet Functions Properties
4 Discrete Wavelet TransformDiscrete and Continuous Wavelet TransformSubband Coding AlgorithmMatrix Interpretation
5 Wavelet Transform ApplicationsDiscontinuity Detection in the ECG SignalImage CompressionImage SegmentationNoise Reduction by Wavelet Shrinkage
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 21 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
DWT DWT and CWT
Discrete Wavelet Transform (DWT)CWT provides us with redundant signal representationDWT - derive by critically sampling the CWT (greatreduction in the number of dilations and shifts)
ψu,s(t)=1√sψ
(t−u
s
)⇒ ψj,k(t)=
1√s j0
ψ
(t−k τ0 s j
0
s j0
)(14)
Sampling on the Dyadic Grids0 =2 ⇒ the scale s =2j
τ0 =1 ⇒ the time translation u = k 2j
t denotes time, j , k are integers
ψj,k(t)=1√2jψ
(t−k2j
2j
)(15)
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 22 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
DWT DWT and CWT
Discrete Wavelet Transform (DWT)CWT provides us with redundant signal representationDWT - derive by critically sampling the CWT (greatreduction in the number of dilations and shifts)
ψu,s(t)=1√sψ
(t−u
s
)⇒ ψj,k(t)=
1√s j0
ψ
(t−k τ0 s j
0
s j0
)(14)
Sampling on the Dyadic Grids0 =2 ⇒ the scale s =2j
τ0 =1 ⇒ the time translation u = k 2j
t denotes time, j , k are integers
ψj,k(t)=1√2jψ
(t−k2j
2j
)(15)
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 22 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
DWT DWT and CWT
Discrete & Continuous Wavelet Transform
50 100 150 200 250 300 350 400 450 500
2000
2500
3000
3500
(a) ECG SIGNAL
(b) DISCRETE WAVELET TRANSF.: ABSOLUTE COEFFICIENTS
scal
e
50 100 150 200 250 300 350 400 450 500
16
8
4
2
1
20
40
60
(c) CONTINUOUS WAVELET TRANSF.: ABSOLUTE COEFFICIENTS
time
scal
e
50 100 150 200 250 300 350 400 450 500123456789
10111213141516
10
20
30
40
50
60
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 23 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
DWT Subband Coding
Wavelet and Scaling Filters
200 400 600
−4
−2
0
2
4
6x 10
−3 (a) Db2 WAVELET FCN
time0 0.1 0.2 0.3 0.4
0
0.2
0.4
0.6
0.8
1
(b) Db2 WAVELET FCN: Freq. Response
Mag
nitu
de
ω/2π
200 400 600
0
2
4
x 10−3 (c) Db2 SCALING FCN
time0 0.1 0.2 0.3 0.4
0
0.2
0.4
0.6
0.8
1
(d) Db2 SCALING FCN: Freq. Response
Mag
nitu
de
ω/2π
The spectrum of the wavelet function corresponds to ahigh-pass filter (or a band-pass filter for higher levels)The spectrum of the scaling function corresponds to alow-pass filter
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 24 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
DWT Subband Coding
Wavelet and Scaling Filters
200 400 600
−4
−2
0
2
4
6x 10
−3 (a) Db2 WAVELET FCN
time0 0.1 0.2 0.3 0.4
0
0.2
0.4
0.6
0.8
1
(b) Db2 WAVELET FCN: Freq. Response
Mag
nitu
de
ω/2π
200 400 600
0
2
4
x 10−3 (c) Db2 SCALING FCN
time0 0.1 0.2 0.3 0.4
0
0.2
0.4
0.6
0.8
1
(d) Db2 SCALING FCN: Freq. Response
Mag
nitu
de
ω/2π
The spectrum of the wavelet function corresponds to ahigh-pass filter (or a band-pass filter for higher levels)The spectrum of the scaling function corresponds to alow-pass filter
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 24 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
DWT Subband Coding
Subband Coding Algorithm
ld[n]
hd[n]
2
2
ld[n]
hd[n]
2
2
ld[n]
hd[n]
2
2
1D DWT
LEVEL 1 LEVEL 2 LEVEL 3
originalsignal x[n]
detailcoefficients
D1
detailcoefficients
D2
detailcoefficients
D3
approximationcoefficients
A3
CWT versus DWTCorrelationDilating thewavelets
ConvolutionKeeping filters the same anddown-sampling the previous leveloutput by 2
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 25 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
DWT Subband Coding
Subband Coding Algorithm
ld[n]
hd[n]
2
2
ld[n]
hd[n]
2
2
ld[n]
hd[n]
2
2
1D DWT
LEVEL 1 LEVEL 2 LEVEL 3
originalsignal x[n]
detailcoefficients
D1
detailcoefficients
D2
detailcoefficients
D3
approximationcoefficients
A3
CWT versus DWTCorrelationDilating thewavelets
ConvolutionKeeping filters the same anddown-sampling the previous leveloutput by 2
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 25 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
DWT Subband Coding
Convolution and Downsampling by 2The taps of the high-pass filter hd and the low-pass ldfilter are derived from the wavelet and the scaling function,resp., of a chosen family (e.g. Daubechies, symlets, etc.)
Convolution and Downsampling by 2Approximation coefficients of the first level
A1[n]=
∞∑k=−∞
ld [k] x [2n − k] (16)
Detail coefficients of the first level
D1[n]=
∞∑k=−∞
hd [k] x [2n − k] (17)
The j-th level Aj [n]=
∞∑k=−∞
ld [k] Aj−1[2n − k] (18)
Dj [n]=
∞∑k=−∞
hd [k] Aj−1[2n − k] (19)
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 26 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
DWT Subband Coding
Convolution and Downsampling by 2The taps of the high-pass filter hd and the low-pass ldfilter are derived from the wavelet and the scaling function,resp., of a chosen family (e.g. Daubechies, symlets, etc.)
Convolution and Downsampling by 2Approximation coefficients of the first level
A1[n]=
∞∑k=−∞
ld [k] x [2n − k] (16)
Detail coefficients of the first level
D1[n]=
∞∑k=−∞
hd [k] x [2n − k] (17)
The j-th level Aj [n]=
∞∑k=−∞
ld [k] Aj−1[2n − k] (18)
Dj [n]=
∞∑k=−∞
hd [k] Aj−1[2n − k] (19)
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 26 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
DWT Subband Coding
Convolution and Downsampling by 2The taps of the high-pass filter hd and the low-pass ldfilter are derived from the wavelet and the scaling function,resp., of a chosen family (e.g. Daubechies, symlets, etc.)
Convolution and Downsampling by 2Approximation coefficients of the first level
A1[n]=
∞∑k=−∞
ld [k] x [2n − k] (16)
Detail coefficients of the first level
D1[n]=
∞∑k=−∞
hd [k] x [2n − k] (17)
The j-th level Aj [n]=
∞∑k=−∞
ld [k] Aj−1[2n − k] (18)
Dj [n]=
∞∑k=−∞
hd [k] Aj−1[2n − k] (19)
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 26 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
DWT Subband Coding
Subband Coding & the Frequency Spectrum
Dilated by 2 ⇒ the spectrum is compressed and shiftedFinite number of dilations ⇒ advantageous to use alow-pass filter - derived from the scaling function φ(t)(the counter part of the wavelet filter at each level -creating a filter bank)
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 27 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
DWT Subband Coding
Subband Coding & the Frequency Spectrum
Dilated by 2 ⇒ the spectrum is compressed and shiftedFinite number of dilations ⇒ advantageous to use alow-pass filter - derived from the scaling function φ(t)(the counter part of the wavelet filter at each level -creating a filter bank)
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 27 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
DWT Matrix Interpretation
DWT MatrixThe Haar filters
ld =1/√2 · [1, 1] (20)
hd =1/√2 · [1, −1] (21)
DWT matrix for the Haar filters
W =1√2·
1 1 0 0 . . . 0 0−1 1 0 0 . . . 0 00 0 1 1 . . . 0 00 0 −1 1 . . . 0 0...
......
.... . .
......
0 0 0 0 . . . 1 10 0 0 0 . . . −1 1
(22)
W includes both convolution and down-sampling by 2(the filters are shifted by 2 samples)W is orthonormal ⇒ W ·WT = Iwhere I stands for the identity matrix
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 28 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
DWT Matrix Interpretation
DWT MatrixThe Haar filters
ld =1/√2 · [1, 1] (20)
hd =1/√2 · [1, −1] (21)
DWT matrix for the Haar filters
W =1√2·
1 1 0 0 . . . 0 0−1 1 0 0 . . . 0 00 0 1 1 . . . 0 00 0 −1 1 . . . 0 0...
......
.... . .
......
0 0 0 0 . . . 1 10 0 0 0 . . . −1 1
(22)
W includes both convolution and down-sampling by 2(the filters are shifted by 2 samples)W is orthonormal ⇒ W ·WT = Iwhere I stands for the identity matrix
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 28 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
DWT Matrix Interpretation
DWT of Signal x Using the Haar Filters
1√2
1 1 0 0 . . . 0 0−1 1 0 0 . . . 0 00 0 1 1 . . . 0 00 0 −1 1 . . . 0 0...
......
.... . .
......
0 0 0 0 . . . 1 10 0 0 0 . . . −1 1
·
x(0)x(1)x(2)x(3)...
x(N−1)x(N)
=
A1(0)D1(0)A1(1)D1(1)
...A1( N
2 −1)D1( N
2 −1)
DWT Decomposition and ReconstructionSignal decomposition
W · x = w (23)Signal reconstruction
x = W−1 ·w (24)Signal reconstruction for the orthonormal DWT
x = WT ·w (25)
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 29 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
DWT Matrix Interpretation
DWT of Signal x Using the Haar Filters
1√2
1 1 0 0 . . . 0 0−1 1 0 0 . . . 0 00 0 1 1 . . . 0 00 0 −1 1 . . . 0 0...
......
.... . .
......
0 0 0 0 . . . 1 10 0 0 0 . . . −1 1
·
x(0)x(1)x(2)x(3)...
x(N−1)x(N)
=
A1(0)D1(0)A1(1)D1(1)
...A1( N
2 −1)D1( N
2 −1)
DWT Decomposition and ReconstructionSignal decomposition
W · x = w (23)Signal reconstruction
x = W−1 ·w (24)Signal reconstruction for the orthonormal DWT
x = WT ·w (25)
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 29 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
WT Applications
Table of Contents
1 Introduction to Wavelet Transform
2 Fourier TransformContinuous Fourier TransformShort-Time Fourier Transform (STFT)
3 Wavelet TransformMulti-Resolution Analysis (MRA)Continuous Wavelet Transform (CWT)Wavelet Functions Properties
4 Discrete Wavelet TransformDiscrete and Continuous Wavelet TransformSubband Coding AlgorithmMatrix Interpretation
5 Wavelet Transform ApplicationsDiscontinuity Detection in the ECG SignalImage CompressionImage SegmentationNoise Reduction by Wavelet Shrinkage
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 30 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
WT Applications Discontinuity Detection
Discontinuity Detection in the ECG Signal
ld[n]
hd[n]
2
2
ld[n]
hd[n]
2
2
ld[n]
hd[n]
2
2
(a) 1−D DWT
signál x[n]
detaily D
1
detaily D
2
detaily D
3
aproximace A
3
nízkofrekvenènípropust
vysokofrekvenènípropust
0 100 200 300 400 500
(b) EKG SIGNÁL
time
(c) DB4: ABS. DETAILNÍ KOEFICIENTY
time100 200 300 400 500
Nespojitost
Nespojitost
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 31 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
WT Applications Image Compression
Image Compression
The Parseval theorem: the energy εx conveyed in thesignal x equals the energy of the coefficients w obtainedthrough an orthonormal transform
εx =N−1∑n=0|xn|2 =
N−1∑k=0|wk |2 (26)
(a) ORIGINAL IMAGE (b) DECOMPOSED IMAGE
0.8%
0.7% 0.1%
94.2% 2%
2% 0.4%
(c) COMPRESSED IMAGE : 94.2%
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 32 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
WT Applications Image Segmentation
Magnetic Resonance Image Segmentation
1 Image preprocessing2 Watershed transform3 DWT features extraction4 Features classification using a neural network
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 33 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
WT Applications Noise Reduction
Noise Reduction by Wavelet Shrinkage
NoisyImage
NoisyImage
WaveletAnalysisWaveletAnalysisWaveletAnalysis
Image
HiHi1HiLo1
LoHi1LoLo1
HiHi2HiLo2
LoHi2LoLo2
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
Thresholding procedure type? Threshold level?
Wavelet SynthesisWavelet SynthesisWavelet Synthesis
Image
LoLo2
LoLo1
LoHi1HiLo1
HiHi1LoHi2HiLo2
HiHi2
Wavelet function? Number of levels?Wavelet function? Number of levels?
DenoisedImage
DenoisedImage
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 34 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
WT Applications Noise Reduction
Noise Reduction by Wavelet Shrinkage
NoisyImage
NoisyImage
WaveletAnalysis
WaveletAnalysisWaveletAnalysis
Image
HiHi1HiLo1
LoHi1LoLo1
HiHi2HiLo2
LoHi2LoLo2
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
Thresholding procedure type? Threshold level?
Wavelet SynthesisWavelet SynthesisWavelet Synthesis
Image
LoLo2
LoLo1
LoHi1HiLo1
HiHi1LoHi2HiLo2
HiHi2
Wavelet function? Number of levels?Wavelet function? Number of levels?
DenoisedImage
DenoisedImage
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 34 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
WT Applications Noise Reduction
Noise Reduction by Wavelet Shrinkage
NoisyImage
NoisyImage
WaveletAnalysis
WaveletAnalysisWaveletAnalysis
Image
HiHi1HiLo1
LoHi1LoLo1
HiHi2HiLo2
LoHi2LoLo2
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
Thresholding procedure type? Threshold level?
Wavelet SynthesisWavelet SynthesisWavelet Synthesis
Image
LoLo2
LoLo1
LoHi1HiLo1
HiHi1LoHi2HiLo2
HiHi2
Wavelet function? Number of levels?Wavelet function? Number of levels?
DenoisedImage
DenoisedImage
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 34 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
WT Applications Noise Reduction
Noise Reduction by Wavelet Shrinkage
NoisyImage
NoisyImage
WaveletAnalysis
WaveletAnalysisWaveletAnalysis
Image
HiHi1HiLo1
LoHi1LoLo1
HiHi2HiLo2
LoHi2LoLo2
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
Thresholding procedure type? Threshold level?
Wavelet Synthesis
Wavelet SynthesisWavelet Synthesis
Image
LoLo2
LoLo1
LoHi1HiLo1
HiHi1LoHi2HiLo2
HiHi2
Wavelet function? Number of levels?Wavelet function? Number of levels?
DenoisedImage
DenoisedImage
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 34 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
WT Applications Noise Reduction
Noise Reduction by Wavelet Shrinkage
NoisyImageNoisyImage
WaveletAnalysis
WaveletAnalysisWaveletAnalysis
Image
HiHi1HiLo1
LoHi1LoLo1
HiHi2HiLo2
LoHi2LoLo2
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
Thresholding procedure type? Threshold level?
Wavelet Synthesis
Wavelet SynthesisWavelet Synthesis
Image
LoLo2
LoLo1
LoHi1HiLo1
HiHi1LoHi2HiLo2
HiHi2
Wavelet function? Number of levels?Wavelet function? Number of levels?
DenoisedImage
DenoisedImage
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 34 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
WT Applications Noise Reduction
Noise Reduction by Wavelet Shrinkage
NoisyImage
NoisyImage
WaveletAnalysis
WaveletAnalysis
WaveletAnalysis
Image
HiHi1HiLo1
LoHi1LoLo1
HiHi2HiLo2
LoHi2LoLo2
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
Thresholding procedure type? Threshold level?
Wavelet Synthesis
Wavelet Synthesis
Wavelet Synthesis
Image
LoLo2
LoLo1
LoHi1HiLo1
HiHi1LoHi2HiLo2
HiHi2
Wavelet function? Number of levels?
Wavelet function? Number of levels?
DenoisedImage
DenoisedImage
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 34 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
WT Applications Noise Reduction
Noise Reduction by Wavelet Shrinkage
NoisyImage
NoisyImage
WaveletAnalysisWaveletAnalysis
WaveletAnalysis
Image
HiHi1HiLo1
LoHi1LoLo1
HiHi2HiLo2
LoHi2LoLo2
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
Thresholding procedure type? Threshold level?
Wavelet SynthesisWavelet Synthesis
Wavelet Synthesis
Image
LoLo2
LoLo1
LoHi1HiLo1
HiHi1LoHi2HiLo2
HiHi2
Wavelet function? Number of levels?
Wavelet function? Number of levels?
DenoisedImage
DenoisedImage
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 34 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
WT Applications Noise Reduction
Noise Reduction by Wavelet Shrinkage
(a) ORIGINAL (b) WITH ADD. GAUSSIAN NOISE (c) DENOISED
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 35 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
WT Applications Noise Reduction
Noise Reduction by Wavelet Shrinkage
(a) ORIGINAL (b) WITH ADD. GAUSSIAN NOISE (c) DENOISED
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 35 / 36
Wavelet Transform
Eva Hostalkova
Introduction
Fourier TransformContinuous Fourier Transform
STFT
Wavelet TransformMRA
CWT
Wavelets Properties
DWTDWT and CWT
Subband Coding
Matrix Interpretation
WT ApplicationsDiscontinuity Detection
Image Compression
Image Segmentation
Noise Reduction
References
References
Further Reading I
S. Mallat.A Wavelet Tour of Signal Processing.Academic Press, 1998.
G. Strang, T. Nguyen. Wavelets and Filter Banks.Wellesley-Cambridge Press, 1996.
M. Barni. Document and Image Compression.Taylor & Francis, 2001.
R. Polikar. The Wavelet Tutorial.http://users.rowan.edu/ polikar/, Iowa State University, 2001.
C. Valens. A Really Friendly Guide to Wavelets.http://pagesperso-orange.fr/polyvalens/, 2003.
www.wavelet.org
The MathWorks, Inc. Matlab Help.www.mathworks.com 2009.
Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 36 / 36